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 August 28th, 2013, 07:43 PM #1 Newbie   Joined: Aug 2013 Posts: 7 Thanks: 0 How to show the standard distance in R^n is a metric space How do I show that the standard distance in R^n is a metric space? $d(x,y)= || x - y ||$ is a metric on $\mathbb{R}^n$? I know that I need to show the definition of a metric to prove this like symmetry, triangle inequality, and non-negative but I do not know how exactly to do that? It seems obvious but I don't know how to formally prove it.
August 29th, 2013, 02:43 PM   #2
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Re: How to show the standard distance in R^n is a metric spa

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 Originally Posted by jugger3 How do I show that the standard distance in R^n is a metric space? $d(x,y)= || x - y ||$ is a metric on $\mathbb{R}^n$? I know that I need to show the definition of a metric to prove this like symmetry, triangle inequality, and non-negative but I do not know how exactly to do that? It seems obvious but I don't know how to formally prove it.
First what is the definition of ||x- y|| in $\mathbb{R}^n$? As I recall it involves a square root! That pretty much takes care of "non-negative", doesn't it? And if there is such a thing as $(x_i- y_i)^2$ in that kind of clears "symmetric". The hard part is the "triangle inequality". But I'll bet you knew that.

 August 29th, 2013, 05:24 PM #3 Newbie   Joined: Aug 2013 Posts: 7 Thanks: 0 Re: How to show the standard distance in R^n is a metric spa Thanks HallsofIvy! I should make a better effort to always look back at the definition.

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